13.1 Absolute and relative extrema

Absolute and relative extrema help us find the highest and lowest points of functions. They're crucial for understanding a function's behavior and solving real-world problems involving optimization.

Absolute extrema are the overall highest and lowest points, while relative extrema are local peaks and valleys. Finding these points involves analyzing critical points and endpoints, giving us valuable insights into function behavior.

Absolute and Relative Extrema

Absolute vs relative extrema

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• Absolute extrema refer to the overall maximum and minimum values of a function within a specified domain
  • Absolute maximum is the highest point on the graph of the function within the domain (global maximum)
  • Absolute minimum is the lowest point on the graph of the function within the domain (global minimum)
• Relative extrema are the local maximum and minimum values of a function within a specified domain
  • Relative maximum (local maximum) is a point where the function value is greater than or equal to the function values at nearby points
  • Relative minimum (local minimum) is a point where the function value is less than or equal to the function values at nearby points

Existence of extrema

• A function has an absolute maximum if there exists a point $x_{max}$ in the domain such that $f(x_{max}) \geq f(x)$ for all $x$ in the domain
• A function has an absolute minimum if there exists a point $x_{min}$ in the domain such that $f(x_{min}) \leq f(x)$ for all $x$ in the domain
• To determine if a function has an absolute maximum or minimum:
  1. Evaluate the function at the endpoints of a closed interval domain ($a$ and $b$ for interval $[a, b]$)
  2. Find the critical points of the function within the domain by setting the first derivative equal to zero or identifying points where the derivative is undefined
  3. Evaluate the function at these critical points
  4. Compare the function values at the endpoints and critical points to identify the absolute extrema
• If the domain is an open interval ($a < x < b$) or the entire real line, the function may not have absolute extrema

Finding extrema of functions

• To find relative extrema:
  1. Determine the critical points of the function by setting the first derivative equal to zero or identifying points where the derivative is undefined
  2. Evaluate the function at each critical point
  3. Test points on either side of the critical points to classify them as relative maxima, relative minima, or neither (saddle points)
• To find absolute extrema on a closed interval $[a, b]$:
  1. Evaluate the function at the endpoints $a$ and $b$
  2. Find the critical points of the function within the interval and evaluate the function at these points
  3. Compare the function values at the endpoints and critical points to identify the absolute extrema

Examples of extrema types

• Consider the function $f(x) = x^3 - 3x$ on the interval $[-2, 2]$
  • The relative extrema are at $x = -1$ (relative maximum) and $x = 1$ (relative minimum)
  • The absolute extrema are at $x = -2$ (absolute maximum) and $x = 1$ (absolute minimum)
• Another example is the function $g(x) = \sin(x)$ on the interval $[0, 2\pi]$
  • The relative extrema occur at multiples of $\pi$
    • Relative maxima at odd multiples of $\pi/2$ ($\pi/2$ and $3\pi/2$)
    • Relative minima at even multiples of $\pi/2$ ($0$, $\pi$, and $2\pi$)
  • The absolute maximum is at $x = \pi/2$, and the absolute minimum is at $x = 3\pi/2$
• These examples demonstrate that relative extrema are local maximum and minimum points, while absolute extrema are the overall maximum and minimum points within the given domain